Local Duality Theorem Research Articles

An implementation of the Suwa method for computing first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations

The Suwa method for computing versal unfoldings of holomorphic singular foliations is considered from the point of view of computational complex analysis. Based on the theory of Grothendieck local duality on residues, an effective algorithm of computing a first order infinitesimal versal unfoldings of codimension one complex analytic singular foliations is obtained. As an application of our approach, we give an effective method for computing universal unfoldings of germs of meromorphic functions.

Generalized local duality, canonical modules, and prescribed bound on projective dimension

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.

An effective method for computing Grothendieck point residue mappings

Grothendieck point residue is considered in the context of computational complex analysis. A new effective method is proposed for computing Grothendieck point residue mappings and residues. Basic ideas of our approach are the use of Grothendieck local duality and a transformation law for local cohomology classes. A new tool is devised for efficiency to solve the extended ideal membership problems in local rings. The resulting algorithms are described with an example to illustrate them. An extension of the proposed method to parametric cases is also discussed as an application.

An integral model of the perfectoid modular curve

We construct an integral model of the perfectoid modular curve. Studying this object, we prove some vanishing results for the coherent cohomology at perfectoid level. We use a local duality theorem at finite level to compute duals for the coherent cohomology of the integral perfectoid curve. Specializing to the structural sheaf, we can describe the dual of the completed cohomology as the inverse limit of the integral cusp forms of weight 2 and trace maps.

Self-duality of the local cohomology of the Jacobian ring and Gherardelli’s theorem

We prove that the 0-th local cohomology of the jacobian ring of a projective hypersurface with isolated singularities has a nice interpretation it in the context of linkage theory. Roughly speaking, it represents a measure of the failure of Gherardelli’s theorem for the corresponding graded modules. This leads us to a different and characteristic free proof of its self-duality, which turns out to be an easy consequence of Grothendieck’s local duality theorem.

Explicit versions of the local duality theorem in Cn

We consider versions of the local duality theorem in Cn. We show that there exist canonical pairings in these versions of the duality theorem which can be expressed explicitly in terms of residues of Grothendieck, or in terms of residue currents of Coleff–Herrera and Andersson–Wulcan, and we give several different proofs of non-degeneracy of the pairings. One of the proofs of non-degeneracy uses the theory of linkage, and conversely, we can use the non-degeneracy to obtain results about linkage for modules. We also discuss a variant of such pairings based on residues considered by Passare, Lejeune-Jalabert and Lundqvist.

Relative Canonical Modules and Some Duality Results

Let (R, m) be a relative Cohen–Macaulay local ring with respect to an ideal a of R and set c to be ht a. We investigate some properties of the Matlis dual of the R-module [Formula: see text], and we show that such modules behave like canonical modules over Cohen–Macaulay local rings. Moreover, we provide some duality and equivalence results with respect to the module [Formula: see text], and these results lead us to achieve generalizations of some known results, such as the local duality theorem, which have been provided over a Cohen–Macaulay local ring admiting a canonical R-module.

Injective DG-modules over non-positive DG-rings

Let A be an associative non-positive differential graded ring. In this paper we make a detailed study of a category Inj(A) of left DG-modules over A which generalizes the category of injective modules over a ring. We give many characterizations of this category, generalizing the theory of injective modules, and prove a derived version of the Bass–Papp theorem: the category Inj(A) is closed in the derived category D(A) under arbitrary direct sums if and only if the ring H0(A) is left noetherian and for every i<0 the left H0(A)-module Hi(A) is finitely generated. Specializing further to the case of commutative noetherian DG-rings, we generalize the Matlis structure theory of injectives to this context. As an application, we obtain a concrete version of Grothendieck's local duality theorem over commutative noetherian local DG-rings.

Local duality in algebra and topology

The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable ∞-category C together with a collection of compact objects K⊂C we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring A and a suitable ideal in A, we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme X implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman.As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts.

A local duality principle in derived categories of commutative Noetherian rings

Let R be a commutative Noetherian ring. We introduce the notion of colocalization functors γW with supports in arbitrary subsets W of SpecR. If W is a specialization-closed subset, then γW coincides with the right derived functor RΓW of the section functor ΓW with support in W. We prove that the local duality theorem and the vanishing theorem of Grothendieck type hold for γW with W being an arbitrary subset.

Local duality for structured ring spectra

We use the abstract framework constructed in our earlier paper [8] to study local duality for Noetherian E∞-ring spectra. In particular, we compute the local cohomology of relative dualizing modules for finite morphisms of ring spectra, thereby generalizing the local duality theorem of Benson and Greenlees. We then explain how our results apply to the modular representation theory of compact Lie groups and finite group schemes, which recovers the theory previously developed by Benson, Iyengar, Krause, and Pevtsova.

A differential operator for integrating one-loop scattering equations

We propose a differential operator for computing the residues associated with a class of meromorphic n-forms that frequently appear in the Cachazo-He-Yuan form of the scattering amplitudes. This differential operator is conjectured to be uniquely determined by the local duality theorem and the intersection number of the divisors in the n-form. We use the operator to evaluate the one-loop integrand of Yang-Mills theory from their generalized CHY formulae. The method can reduce the complexity of the calculation. In addition, the expression for the 1-loop four-point Yang-Mills integrand obtained in our approach has a clear correspondence with the Q-cut results.

Notes on local cohomology and duality

We provide a formula (see Theorem 1.5) for the Matlis dual of the injective hull of R/p where p is a one dimensional prime ideal in a local complete Gorenstein domain (R,m). This is related to results of Enochs and Xu (see [4] and [5]). We prove a certain ‘dual’ version of the Hartshorne–Lichtenbaum vanishing (see Theorem 2.2). We prove a generalization of local duality to cohomologically complete intersection ideals I in the sense that for I=m we get back the classical Local Duality Theorem. We determine the exact class of modules to which a characterization of cohomologically complete intersection from [7] generalizes naturally (see Theorem 4.4).

Local duality theorem for q-ary 1-perfect codes

In this paper, we derive the relationship between local weight enumerator of q-ary 1-perfect code in a face and that in the orthogonal face. As an application of our result, we compute the local weight enumerators of a shortened, doubly-shortened, and triply-shortened q-ary 1-perfect code.

Logarithmic differential forms, torsion differentials and residue†

This article deals with the theory of logarithmic differential forms and with some of its less known applications, developed by the author in the past few years, in complex analysis, topology and geometry of singular varieties and in the theory of differential equations. At first, some relations between properties of logarithmic differential forms and torsion differentials on singular hypersurfaces are studied. Then a natural extension of the classical Poincaré residue in the case of singular hypersurfaces, and a variant of the Grothendieck local duality for non-isolated non-normal reduced hypersurface singularities are described in detail. In conclusion a new method for computing the index of vector fields on hypersurfaces with arbitrary singularities, and some applications to the theory of Pfaffian systems of differential equations of Fuchsian type are discussed.

On the structure theory of the Iwasawa algebra of a p-adic Lie group

This paper is motivated by the question whether there is a nice structure theory of finitely generated modules over the Iwasawa algebra, i.e. the completed group algebra, 6 of a p-adic analytic group G. For G without any p-torsion element we prove that 6 is an Auslander regular ring. This result enables us to give a good definition of the notion of a pseudo-null 6-module. This is classical when G=Êkp for some integer kS1, but was previously unknown in the non-commutative case. Then the category of 6-modules up to pseudo-isomorphisms is studied and we obtain a weak structure theorem for the Êp-torsion part of a finitely generated 6-module. We also prove a local duality theorem and a version of Auslander-Buchsbaum equality. The arithmetic applications to the Iwasawa theory of abelian varieties are published elsewhere.

Rings with Auslander Dualizing Complexes

A ring with an Auslander dualizing complex is a generalization of an Auslander–Gorenstein ring. We show that many results which hold for Auslander–Gorenstein rings also hold in the more general setting. On the other hand we give criteria for existence of Auslander dualizing complexes which show these occur quite frequently. The most powerful tool we use is the Local Duality Theorem for connected graded algebras over a field. Filtrations allow the transfer of results to nongraded algebras. We also prove some results of a categorical nature, most notably the functoriality of rigid dualizing complexes.

A local duality theorem for categories of modules

Let Λ be an associative ring with identity and let $$_\Lambda \mathfrak{M}$$ be the category of left unitary Λ-modules. A subcateqory $$\mathcal{M}$$ of the category $$_\Lambda \mathfrak{M}$$ is said to be small if the pairwise nonisomorphic objects of $$\mathcal{M}$$ form a set. The main result of this paper consists of the fact that for every small full subcategory $$\mathcal{M}$$ , there exists a ring Γ such that $$\mathcal{M}$$ is dual to a small full subcategory of the category $$_\Gamma \mathfrak{M}$$ . Some applications of this result are indicated. Bibliography: 3 titles.

Local homology and cohomology on schemes

We present a sheafified derived-category generalization of Greenlees-May duality (a far-reaching generalization of Grothendieck's local duality theorem): for a quasi-compact separated scheme X and a “proregular” subscheme Z-for example, any separated noetherian scheme and any closed subscheme-there is a sort of adjointness between local cohomology supported in Z and left-derived completion along Z. In particular, left-derived completion can be identified with local homology, i.e., the homology of R H em •( RΛ Z Q X−). Generalizations of a number of duality theorems scattered about the literature result: the Peskine-Szpiro duality sequence (generalizing local duality), the Warwick Duality theorem of Greenlees, the Affine Duality theorem of Hartshorne. Using Grothendieck Duality, we also get a generalization of a Formal Duality theorem of Hartshorne, and of a related local-global duality theorem. In a sequel we will develop the latter results further, to study Grothendieck duality and residues on formal schemes.

NORMAL FORMS OF ONE-DIMENSIONAL QUASIHOMOGENEOUS COMPLETE INTERSECTIONS

In this paper the author presents an approach to the problem of classifying quasihomogeneous singularities, based on the use of simple properties of deformation theories of such singularities. By means of Grothendieck local duality the Poincare series of the space of the first cotangent functor T' of a one-dimensional singularity is computed. Lists of normal forms and monomial bases of the spaces of T' are given for one-dimensional quasihomogeneous complete intersections with inner modality 0 and 1, and also with Milnor number less than seventeen. An adjacency diagram is constructed for all singularities that have been found. Bibliography: 33 titles.